W965: rung sixteen is strictly dominated — same lane cost, more range, more values, decided by the grids alone

[gHashTag]

The last row of the ladder marked "unmeasured" on all three axes is closed — and closing it needed no training and no synthesis.

Both versions, labelled by version

LADDER (v1-research) is TNFFormat(4, 9) at 17 bits; get_ladder(DEFAULT) (v2-spec) is TNFFormat(4, 11) at 19 bits. After #668 no rung here is taken by its name again.

The grids decide it

Factoring every grid value as |v| = odd · 2^s · u (a float is dyadic, so this is exact) gives the pair that #669 showed fixes lane cost at equal width:

format	bits	distinct values	binades	odd	max shift	aligned bus
TNF16 v1-research (4,9)	17	129 025	127.00	10	135	290
fp17 e7m9 (range-matched)	17	131 071	136.00	10	135	290
TNF16 v2-spec (4,11)	19	516 097	127.00	12	137	298
fp19 e7m11 (range-matched)	19	524 287	138.00	12	137	298

The structural pairs are identical — 10/135 and 12/137 — so the lane cost, the aligned bus and the accumulator are identical. And the ordinary float dominates on both remaining axes: more distinct values, and more range. The φ-lattice loses about 1.6 % of its code space to points that are unrepresentable or duplicated.

This is not parity. It is strict domination, and it is a property of the grids rather than of any experiment.

The alternative framing is no kinder

A width-matched float with a conventional exponent (fp17 e6m10) spans 73 binades on a 166-bit bus against the rung's 290. So at these widths the choice against the φ-lattice is between the same cost with more range and more values, or far less cost with less range. The rung is on neither frontier.

Scope, stated

The cost equality is a prediction from #669's law, which was validated by synthesis at six and ten bits — not a measurement at seventeen and nineteen, where a case table over 2¹⁹ codes is not buildable. The value counts and structural parameters are exact. A synthesis check at these widths needs a structural decoder.

The ladder, as far as it has been measured

rung	area	accuracy	stability
4 (6 bits)	parity at matched range	no advantage	no advantage under the deployed recipe
8 (10 bits)	parity (−0.9 % / +1.1 % in two metrics)	parity, three recipes	no failures
16 (17/19 bits)	strictly dominated	—	—

python3 research/arxiv_tnf/verify_numbers.py

185 derived checks, all passing. Related: #644, #664, #668, #670, #671.

🤖 Generated with Claude Code